Effects of mis-specification of seasonal cointegrating ranks: An - - PowerPoint PPT Presentation

SLIDE 1

10-12 May 2006

Effects of mis-specification of seasonal cointegrating ranks: An empirical study

Byeongchan Seong Sinsup Cho

• S. Y. Hwang

Sung K. Ahn

Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting

SLIDE 2

Effects of mis-specification of seasonal cointegrating ranks: An empirical study1

Byeongchan Seong

Pohang University of Science and Technology

Sinsup Cho

Seoul National University

• S. Y. Hwang

Sookmyung Women’s University

Sung K. Ahn

Washington State University

1 Byeongchan Seong’s research was supported by the Post-doctoral Fellowship Program of

Korea Science & Engineering Foundation (KOSEF). The research of Sinsup Cho, S. Y. Hwang, and Sung K. Ahn was supported by the Korea Research Foundation Grant (KRF- 2005-070-C00022) funded by the Korean Government (MOEHRD).

SLIDE 3

Co-integration (Engle & Granger, 1987)

An m-dimensional I(d) process

t

y is co- integrated, if there exists a vector β such that

t

y β′ is an I(b) process, d b < , denoted by CI(d, d-b). Typically, processes are CI(1, 0), i.e., d=1 & b=0. The number of linearly independent vectors is called the co-integrating rank, denoted by r. “Disappearance” of the non-stationarity, or unit root in

t

y β′ is attributable to the common feature (Engle & Kozicki, 1993), more specifically called, common trend in some or all of the elements of

t

y .

SLIDE 4

Seasonal Co-integration

(Hylleberg, Engle, Granger & Yoo,1990) A seasonal process

t

y with period s is seasonally co-integrated at frequency f, if there exists a vector β such that

t

y β′ does not have the seasonal unit root

θ i

e corresponding to the frequency f, f π θ 2 = . Since seasonal unit roots exist in conjugate pairs, there exists polynomial co-integrating vector L

I R

β β + such that

t I R

L y β β ) ( ′ + does not have the seasonal unit root

θ i

e±

SLIDE 5

The characteristics of (seasonal) co- integration are concentrated in the error correction terms through the reduced ranks of the coefficient matrices.

t t t

C L L ε y y + = − Φ

−1 *

) 1 )( (

t

L L y ) 1 )( (

4 *

− Φ

1 2 1 1 − − +

=

t t

C C v u

t t t

C C ε w w + + +

− − 2 4 1 3

where

1 2 1

) 1 )( 1 (

− −

+ + =

t t

L L y u ,

1 2 1

) 1 )( 1 (

− −

+ − =

t t

L L y v , and

1 2 1

) 1 (

− −

− =

t t

L y w Statistical inference of co-integration involves reduced rank estimation in the error correction representation of the vector autoregressive model.

SLIDE 6

Multivariate Regression Model

ε x x x x z + + + + =

4 4 3 3 2 2 1 1

C C C C Estimation:

• Regression of on

, , , and simultaneously. z

1

x

2

x

3

x

4

x

• Regression of on

for each z

j

x 4 , , 1 K = j if the ’s are uncorrelated.

j

x

• Partial regression of on

adjusted for z

j

x

k

x , j k ≠ for each 4 , , 1 K = j .

SLIDE 7

Partial regression is especially useful if one

• f the

’s, say is of reduced rank. To estimate with the reduced rank structure imposed:

j

C

1

C

1

C

• Regress z on

, , and and get the residual ;

2

x

3

x

4

x

z

r

• Regress
• n

, , and and get the residual ;

1

x

2

x

3

x

4

x

1

r

• Reduded-rank regress on ,

z

r

1

r as in Anderson (1951). In co-integration analysis

t t t

C L L ε y y + = − Φ

−1 *

) 1 )( (

• Regress

t

L y ) 1 ( −

• n lagged

t

L y ) 1 ( − and get the residual ;

y

r

• Regress

1 − t

y

• n on lagged

t

L y ) 1 ( − and get the residual ;

1

r

• Reduced-rank regress
• n ,

y

r

1

r as in Johansen (1988).

SLIDE 8

In seasonal co-integration analysis, more than

• ne

’s in

j

C ε x θ x θ x x z + + + + =

4 4 3 3 2 2 1 1

) ( ) ( C C C C can be of reduced rank and some of the ’s are dependent on the common parameter vector.

j

C If and are of reduced rank and and depend on θ, then:

1

C

2

C

3

C

4

C Since the adjustment for , , and is based on the full rank regression, partial reduced-rank regression of z on is affected by over-specification of the rank of ;

2

x

3

x

4

x

1

x

2

C Since the adjustment for , , and is based on the full rank regression, the dependence between and is ignored in partial (reduced-rank) regression of

• n

.

1

x

2

x

3

x

3

C

4

C z

4

x

SLIDE 9

Seasonal Co-integration

1 1 1 4 *

) 1 )( (

−

′ = − Φ

t R R t

β α L L u y

1 2 2 −

′ +

t R Rβ

α v

1 3 3 3 3

) (

−

′ + ′ +

t R I I R

β α β α w

t t I I R R

β α β α ε w + ′ + ′ − +

−2 3 3 3 3

) ( Lee (1992), Johansen & Schaumburg (1999), and Cubadda (2001) use partial reduced rank regression exploiting asymptotic zero correlations:

• Lee assumes

3 = I

α and

3 = I

β ;

• J&S uses the “switching” algorithm to

estimate , ,

R

α3

R

β3

I

α3 , and

I

β3 ;

• Cubadda, in essence, estimates

, ,

R

α3

R

β3

I

α3 , and

I

β3 based on partial regression of

t

L y ) 1 (

4

−

• n

1 − t

w . These create over-specification problems.

SLIDE 10

Ahn & Reinsel (1994) and Ahn, Cho &Seong (2004) use an iterative scheme that incorporates

• the co-integrating ranks at all the seasonal

frequencies simultaneously and

• the dependency among the coefficient

matrices. But this requires the correct specification of the seasonal co-integrating ranks and is subject to over- and under-specification. (Furthermore, it can be computationally challenging.)

SLIDE 11

Simulation Study

DGP (Ahn & Reinsel, 1994):

1 2 2 1 1 1 4 − −

′ + ′ = −

t t t

L v β α u β α y ) (1

1 3 4 4 3

) (

−

′ + ′ +

t

w β α β α

t t

ε w β α β α + ′ + ′ − +

−2 4 4 3 3

) ( where ) 6 . , 6 . ( ) , (

21 11 1

′ = ′ = α a a , ) 6 . , 4 . ( ) , (

22 12 2

′ − = ′ = α a a , ) 6 . , 6 . ( ) , (

23 13 3

′ − = ′ = α a a , ) 8 . , 4 . ( ) , (

24 14 4

′ − = ′ = α a a , ) 7 . , 1 ( ) , 1 (

1 1

′ − = ′ = β b , ) 3 . , 1 ( ) , 1 (

2 2

′ = ′ = b β , ) 7 . , 1 ( ) , 1 (

3 3

′ = = β b , ) 2 . , ( ) , (

4 4

′ − = = β b ,

SLIDE 12

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Ω =

2

1 ) ( σ ρσ ρσ

t

Cov ε for 5 . , , 5 . − = ρ and 2 , 1 , 5 .

2 =

σ . Series length: 100 Replications: 1000 Nominal size: 0.05 Critical values from Johansen & Schaumburg (1999) and Lee & Siklos (1995) For : =

f

r H vs :

1

>

f

r H for 4 / 1 , 2 / 1 , = f , H is rejected almost all the cases regardless of under or over- specification. For 1 : =

f

r H vs 1 :

1

>

f

r H , the results are summarized below.

SLIDE 13

Table 1. Comparison of the rejection rates of 5% level tests for hypotheses in (6) for the frequency . 2 / 1 = f C.I. ranks ) , , (

2 / 1 4 / 1

r r r ρ

2

σ (0,0,1) (0,1,1) (0,2,1) (1,0,1) (1,1,1) (1,2,1) (2,0,1) (2,1,1) (2,2,1) 0.5 0.084 0.024 0.029 0.037 0.050 0.055 0.035 0.050 0.055 1 0.084 0.037 0.043 0.039 0.058 0.059 0.040 0.059 0.059

• 0.5

2 0.081 0.062 0.070 0.035 0.060 0.061 0.034 0.059 0.061 0.5 0.086 0.026 0.029 0.035 0.056 0.059 0.040 0.057 0.062 1 0.083 0.035 0.038 0.040 0.060 0.065 0.042 0.060 0.065 2 0.088 0.053 0.056 0.041 0.064 0.066 0.040 0.063 0.066 0.5 0.082 0.020 0.025 0.045 0.056 0.058 0.051 0.055 0.057 1 0.075 0.025 0.033 0.040 0.060 0.065 0.044 0.060 0.063 0.5 2 0.079 0.043 0.047 0.043 0.064 0.063 0.046 0.063 0.063

• Significantly larger empirical sizes with

under-specification for f=0 & 1/2.

• Significantly smaller empirical sizes with

under-specification for only one of f=0 & 1/2.

SLIDE 14

Table 1-2. Means and mean squared errors for frequency 1/2 in case of 5 . = ρ and based on 1,000 replications. 2

2 =

σ 4 .

12

− = a 6 .

22 =

a 3 .

2 =

b CI ranks ) , , (

2 / 1 4 / 1

r r r Mean MSE Mean MSE Mean MSE (0,0,1)

• 0.3060

0.0182 0.7821 0.0624 0.2938 0.0002 (0,1,1)

• 0.2391

0.0541 0.6164 0.0566 0.2989 0.0004 (0,2,1)

• 0.2521

0.0462 0.6031 0.0534 0.2977 0.0009 (1,0,1)

• 0.3186

0.0148 0.6789 0.0196 0.2931 0.0002 (1,1,1)

• 0.3621

0.0107 0.5145 0.0443 0.3014 0.0000 (1,2,1)

• 0.3603

0.0102 0.5160 0.0430 0.3014 0.0001 (2,0,1)

• 0.3099

0.0148 0.6701 0.0181 0.2938 0.0001 (2,1,1)

• 0.3600

0.0108 0.5038 0.0463 0.3014 0.0001 (2,2,1)

• 0.3581

0.0104 0.5055 0.0449 0.3014 0.0001

• Serious biases occur with under-

specification for the stationary parameters.

• Biases are not serious with under-

specification for the long-run parameter.

SLIDE 15

Table 2. Comparison of the rejection rates of 5% level tests for hypotheses in (6) for the frequency . = f CI ranks ) , , (

2 / 1 4 / 1

r r r ρ

2

σ (1,0,0) (1,0,1) (1,0,2) (1,1,0) (1,1,1) (1,1,2) (1,2,0) (1,2,1) (1,2,2) 0.5 0.249 0.233 0.236 0.028 0.025 0.025 0.025 0.023 0.025 1 0.262 0.256 0.262 0.030 0.022 0.024 0.031 0.024 0.024

• 0.5

2 0.273 0.276 0.280 0.035 0.025 0.025 0.038 0.026 0.026 0.5 0.318 0.323 0.324 0.028 0.014 0.016 0.030 0.013 0.016 1 0.340 0.345 0.351 0.039 0.018 0.019 0.041 0.019 0.019 2 0.376 0.374 0.373 0.047 0.020 0.020 0.047 0.022 0.023 0.5 0.386 0.401 0.407 0.040 0.013 0.013 0.039 0.013 0.015 1 0.445 0.464 0.467 0.053 0.014 0.014 0.052 0.014 0.014 0.5 2 0.494 0.516 0.526 0.061 0.014 0.014 0.059 0.014 0.014

• Significantly larger empirical sizes with

under-specification.

• Significantly larger empirical sizes with

under-specification for f=1/4 and over- specification for f=1/2.

• Significantly smaller empirical sizes with
• ver-specification for f=1/4 and under-

specification for f=1/2.

SLIDE 16

Table 3. Comparison of the rejection rates of 5% level tests for hypotheses in (6) for the frequency . 4 / 1 = f C.I. ranks ) , , (

2 / 1 4 / 1

r r r ρ

2

σ (0,1,0) (0,1,1) (0,1,2) (1,1,0) (1,1,1) (1,1,2) (2,1,0) (2,1,1) (2,1,2) 0.5 0.093 0.058 0.066 0.019 0.020 0.017 0.018 0.019 0.017 1 0.132 0.100 0.105 0.020 0.018 0.019 0.020 0.017 0.018

• 0.5

2 0.167 0.138 0.143 0.019 0.016 0.016 0.020 0.016 0.016 0.5 0.083 0.048 0.055 0.024 0.025 0.024 0.024 0.025 0.024 1 0.112 0.091 0.094 0.027 0.029 0.030 0.027 0.029 0.031 2 0.137 0.113 0.116 0.030 0.024 0.026 0.030 0.025 0.026 0.5 0.139 0.094 0.095 0.073 0.075 0.079 0.074 0.076 0.080 1 0.165 0.128 0.134 0.075 0.084 0.086 0.076 0.084 0.083 0.5 2 0.199 0.164 0.167 0.078 0.077 0.077 0.079 0.079 0.078

• Significantly larger empirical sizes with

under-specification for both f=0 and 1/4 and for only f=0.

• No significant difference with under-

specification for only f=1/2.

• Significantly larger empirical sizes with

under-specification for f=0 and over- specification for f=1/2.

• No significant difference with over-

specification for f=0 and under- specification for f=1/2.

SLIDE 17

Summary

• Over specification of co-integrating ranks

is acceptable, and so is the partial regression based approach.

• May need to check the validity of the

critical values.

• Further simulation study is needed.
• Theoretical investigation in needed.